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The Maze of Systems Of Equations
Transcript of The Maze of Systems Of Equations
are easy to graph.
Solving by Graphing
The third way to solve systems of equations is by Elimination. Elimination can be used to solve systems of equations by adding terms vertically. Then by adding terms vertically, it will cause one of the variables to be eliminated. In some case, it may be required to multiply one or both equations by a number to enable this method. This method is best used when both equations are in standard form.
Solving by Elimination(with scaling)
In systems of equations, there are only 3 possible solutions: intersecting lines, parallel lines and coinciding lines. Intersecting Lines is where you have one solution, the point where the lines intersect is the final solution. Parallel Lines is when these lines never intersect and since the lines never intersect there is no solution. You
can identify if the system is going to be no solution, because the equations would have same slope, but have different y-intercepts. Coinciding Lines is when the lines are the same. It is easy to identify coinciding lines, because both lines have the same slope and the same y-intercept. Since, the lines are on top of each other
they are infinitely many solutions/ all real numbers.
Hence, the two special cases are parallel
lines and coinciding lines.
Special Cases and how they appear each method
The 4th way to solve systems of equations is by solving with Matrices. This method is also called the Gaussian elimination. In order, to use this matrices method you must master how matrices work. In this method, you will have 2 kinds of matrix, one is called: coefficient matrix because it is made from the coefficients of the variable. The second one is called: the constant matrix because it is created from the constants in the system. Basically, the process is to set up the matrices, find the inverse and then multiply the matrices. To use the matrices method to solve systems of equations, you need to utilize both sides of the equation(left and right). Matrices and elimination are quite similar, but matrices have more aspects to it. Solving with matrices possess some complicated steps compared to the other methods. Therefore, using the matrices method is best used when you have 3 equations in your system and all of them are in standard form.
Solving By Matrices
Introduction to Systems of Equations(The history behind it)
Systems of Equations were developed a long time ago and is still essential in our world today. Systems of Equations are basically, a set of two or more equations that consists of the same variables. Systems of equations are solved using 4 methods, but many people usually use 3 of them. These 3 are: solving by graphing, solving by substitution and solving by elimination. The 4th method is solving with matrices. In solving systems of equations,
you will also encounter special cases. Therefore, as you
try to exit this maze you will learn about all aspects of
systems of equations.
Still confused? Before, we start watch the video for a little more background information on the topic and on the methods we are
going to be looking at.
A little more information
Example of how to Solve by graphing and steps of how to do so:
Example of Special Cases- no solution(parallel lines)
The Labyrinth of Systems Of
Do You Dare to go in?
This a video that summarizes systems of equations. and the methods used to solve.
Solving By Substitution
I forgot to tell
you there is no
leaving .... until... these questions are solved...
Solve these systems using the easiest and quickest method.
So more help to understand more on graphing to solve systems of equations.
So more help to understand more on using substitution to solve systems of equations.
So more help to understand more on using elimination to solve systems of equations.
Example of Special Cases- infinitely many solutions(consecutive lines)
Example of how to Solve by substitution and steps of how to do so:
Example of how to solve by elimination and steps of how to do so:
Another way to solve systems of equations is by using substitution. Many use substitution, because sometimes it is difficult to identify the exact solution to a system using the graphing method. When we are in these cases , it is possible to use algebra to solve the system. Therefore, one of the algebraic methods is substitution. To add on, the goal of substitution is to reduce the system to one equation that includes only one variable. Basically, in solving with substitution, you substitute values or equations from the first equation or second for another variable in the other equation. Thus, substitution works best if one variable is already solved for or if it is easy to solve for a variable.
Always remember: that your solution must be ion coordinate form(x,y)
Hint: graphing will be easier
*This question should be answered in one step*
Example of Solving with Matrices
So the first step is to the two
equations vertically, so then as you
vertically add you will be eliminating a
variable. Usually, if the variables in
both equations don't cancel out, then
you would multiply or scale that or both equation up. With this we don't have to scale, we can
cancel out the y variables.
From adding the two equations
vertically it has made a new equation
that contains only one variable.
You divide, 26 by 13 and 13x by
13, then you have the answer for x.
Your not done, but it is
essential to have the
solution for x, so you can find out y. To do this, you plug in the value of x in to one of the equations and then you get your solution for y.
Always remember: your final solution must be in coordinate from(x,y)
The last step to follow through with is check. In order to check this you plug in the values of x and y into one of the equations or to double check both equations.
Here's a fun video for you to always remember systems of equations and how to solve them. Hope you enjoy! Thanks :)
The first step here is to
substitute the y, since y equals y the equations equal to each other. Also, this step is using the method of substitution.
The next step is to solve the equation
how you would normally solve an equation. As you can see there is no x or y= and there is two different numbers on each side of the equal sign. So would that be the answer?
The answer is no, these numbers are not equal; therefore there is no solution.
On a graph, no solution will be shown as parallel lines because they don't intersect, like the ones with one solution.
From the system given, you should see the possible solution right there, but in order to show work- you must follow along.
The second step is to use substitution. You have to substitute the y, since y equals y the equations equal to each other. Also, this step is using the method of substitution. Hence, you would solve it a usual and get an answer with the same number on each side of the equal sign. What does this mean?
The answer is infinitely many solutions, because the same number is equal to its own number, and all real numbers are equal to its own value, so there are an infinite amount of solutions.
The last step is how to portray this on a graph/ coordinate plane. Since it is the same number equals to the same number, then that means the same line equals to the same line.
Need a little help! Watch these videos they both help a lot in explaining the concept to you!